# Accession Number:

## AD0412936

# Title:

## MONOTONICITY PROPERTIES OF THE POWER FUNCTIONS OF SOME TESTS IN TWO MULTIVARIATE PROBLEMS,

# Descriptive Note:

# Corporate Author:

## COLUMBIA UNIV NEW YORK

# Personal Author(s):

# Report Date:

## 1963-07-10

# Pagination or Media Count:

## 13.0

# Abstract:

Invariant procedures for testing a set of multi variate linear hypotheses in the linear normal model depend on the characteristic roots of a random matrix similarly invariant test pro cedures for testing independence between two sets of normally distributed variates depend on the characteristic roots of another random matrix. In each case the power function of such a test depends on the characteristic roots of a cor responding population matrix as parameters these roots may be regarded as measures of deviation from the hypothesis tested. Sufficient condi tions on the procedure in each case for the power function to be a monotonically increasing func tion of each of the parameters are obtained. The likelihood-ratio test, Lawley-Hotelling trace test, and Roys maximum root test satisfy these conditions. The monotonicity of the power func tion of Roys test has been shown by Roy and Mikhail using a geometrical method. In only the unbiasedness of the maximum root test of independence was proved although the authors claimed to prove the monotonicity property. Author